Aims. The aim of this work is to shed some light on the problem of the formation of carbon stars of R-type from a detailed study of their chemical composition. Methods: We use high-resolution and high signal-to-noise optical spectra of 23 R-type stars (both early- and late-types) selected from the Hipparcos catalogue. The chemical analysis is made using spectral synthesis in LTE and state-of-the-art carbon-rich spherical model atmospheres. We derive their CNO content (including the 12C/13C ratio), average metallicity, lithium, and light (Sr, Y, Zr) and heavy (Ba, La, Nd, Sm) s-element abundances. The observed properties of the stars (galactic distribution, kinematics, binarity, photometry and luminosity) are also discussed. Results: Our analysis shows that late-R stars are carbon stars with identical chemical and observational characteristics as the normal (N-type) AGB carbon stars. The s-element abundance pattern derived can be reproduced by low-mass AGB nucleosynthesis models where the 13C(α, n)16O reaction is the main neutron donor. We confirm the results of the sole previous abundance analysis of early-R stars, namely that they are carbon stars with near solar metallicity showing enhanced nitrogen, low 12C/13C ratios and no s-element enhancements. In addition, we have found that early-R stars have Li abundances larger than expected for post RGB tip giants. We also find that a significant number (~40%) of the early-R stars in our sample are wrongly classified, probably being classical CH stars and normal K giants. Conclusions: On the basis of the chemical analysis, we confirm the previous suggestion that late-R stars are just misclassified N-type carbon stars in the AGB phase of evolution. Their photometric, kinematic, variability and luminosity properties are also compatible with this. In consequence, we suggest that the number of true R stars is considerably lower than previously believed. This alleviates the problem of considering R stars as a frequent stage in the evolution of low-mass stars. We briefly discuss the different scenarios proposed for the formation of early-R stars. The mixing of carbon during an anomalous He-flash is favoured, although no physical mechanism able to trigger that mixing has been found yet. The origin of these stars still remains a mystery.